Optimal. Leaf size=20 \[ \frac {x \left (1-x^8\right )^{3/4}}{x^8-1} \]
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Rubi [A] time = 0.00, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 383} \begin {gather*} -\frac {x}{\sqrt [4]{1-x^8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 383
Rubi steps
\begin {align*} \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx &=-\int \frac {1+x^8}{\left (1-x^8\right )^{5/4}} \, dx\\ &=-\frac {x}{\sqrt [4]{1-x^8}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} -\frac {x}{\sqrt [4]{1-x^8}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \left (1-x^8\right )^{3/4}}{-1+x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 18, normalized size = 0.90 \begin {gather*} \frac {{\left (-x^{8} + 1\right )}^{\frac {3}{4}} x}{x^{8} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (-x^{8} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 13, normalized size = 0.65
method | result | size |
gosper | \(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\) | \(13\) |
trager | \(\frac {x \left (-x^{8}+1\right )^{\frac {3}{4}}}{x^{8}-1}\) | \(19\) |
meijerg | \(-\hypergeom \left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], x^{8}\right ) x -\frac {\hypergeom \left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], x^{8}\right ) x^{9}}{9}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 29, normalized size = 1.45 \begin {gather*} -\frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 12, normalized size = 0.60 \begin {gather*} -\frac {x}{{\left (1-x^8\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + 1}{\sqrt [4]{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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